Distinguishing number and distinguishing index of some operations on graphs
Saeid Alikhani, Samaneh Soltani

TL;DR
This paper investigates how the distinguishing number and index of a graph change under various graph operations such as vertex removal, edge removal, and contractions, providing bounds and inequalities for these changes.
Contribution
It establishes new bounds on the changes in the distinguishing number and index when applying common graph operations, extending understanding of graph symmetries.
Findings
Vertex removal changes are bounded between -1 and D(G).
Edge removal alters D(G) and D'(G) by at most 2 and 2 respectively.
Contractions can decrease D(G) and D'(G) by at most 1, or increase them by up to 3 times their original values.
Abstract
The distinguishing number (index) () of a graph is the least integer such that has an vertex labeling (edge labeling) with labels that is preserved only by a trivial automorphism. We examine the effects on and when is modified by operations on vertex and edge of . Let be a connected graph of order . We show that , where denotes the graph obtained from by removal of a vertex and all edges incident to and these inequalities are true for the distinguishing index. Also we prove that and , where denotes the graph obtained from by simply removing the edge . Finally we consider the vertex contraction and the edge contraction of and prove that the edge contraction decrease the distinguishing number (index) of by…
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